On universal oracle inequalities related to high-dimensional linear models

This paper deals with recovering an unknown vector $\theta$ from the noisy data $Y=A\theta+\sigma\xi$, where $A$ is a known $(m\times n)$-matrix and $\xi$ is a white Gaussian noise. It is assumed that $n$ is large and $A$ may be severely ill-posed. Therefore, in order to estimate $\theta$, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data $Y$. For spectral regularization methods related to the so-called ordered smoothers [see Kneip Ann. Statist. 22 (1994) 835--866], we propose new penalties in the principle of empirical risk minimization. The heuristical idea behind these penalties is related to balancing excess risks. Based on this approach, we derive a sharp oracle inequality controlling the mean square risks of data-driven spectral regularization methods.Comment: Published in at http://dx.doi.org/10.1214/10-AOS803 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Weak localization in a system with a barrier: Dephasing and weak Coulomb blockade

We non-perturbatively analyze the effect of electron-electron interactions on weak localization (WL) in relatively short metallic conductors with a tunnel barrier. We demonstrate that the main effect of interactions is electron dephasing which persists down to T=0 and yields suppression of WL correction to conductance below its non-interacting value. Our results may account for recent observations of low temperature saturation of the electron decoherence time in quantum dots.Comment: published version, 10 page

Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

It was recently shown by Lubetzky and Peres (2016) and by Sardari (2018) that Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the simple random walk in optimal time and have optimal almost-diameter. We prove that this spectral condition can be replaced by a weaker condition, the Sarnak-Xue density of eigenvalues property, to deduce similar results. We show that a family of Schreier graphs of the $SL_{2}\left(\mathbb{F}_{t}\right)$-action on the projective line satisfies the Sarnak-Xue density condition, and hence exhibit the desired properties. To the best of our knowledge, this is the first known example of optimal cutoff and almost-diameter on an explicit family of graphs that are neither random nor Ramanujan

Risk hull method and regularization by projections of ill-posed inverse problems

We study a standard method of regularization by projections of the linear inverse problem $Y=Af+\epsilon$, where $\epsilon$ is a white Gaussian noise, and $A$ is a known compact operator with singular values converging to zero with polynomial decay. The unknown function $f$ is recovered by a projection method using the singular value decomposition of $A$. The bandwidth choice of this projection regularization is governed by a data-driven procedure which is based on the principle of risk hull minimization. We provide nonasymptotic upper bounds for the mean square risk of this method and we show, in particular, that in numerical simulations this approach may substantially improve the classical method of unbiased risk estimation.Comment: Published at http://dx.doi.org/10.1214/009053606000000542 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org